IJMMS 2003:5, 295–304
PII. S0161171203111325
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
CHARACTERIZATIONS OF VECTOR-VALUED WEAKLY
ALMOST PERIODIC FUNCTIONS
CHUANYI ZHANG
Received 10 November 2001
We characterize the weak almost periodicity of a vector-valued, bounded, continuous function. We show that if the range of the function is relatively weakly
compact, then the relative weak compactness of its right orbit is equivalent to
that of its left orbit. At the same time, we give the function some other equivalent
properties.
2000 Mathematics Subject Classification: 43A60, 43A07.
1. Introduction. Let S be a semitopological semigroup, let ᐄ be a Banach
space, and let Ꮿ(S, ᐄ) be the space of bounded continuous functions from S
to ᐄ with supremum norm. Let f ∈ Ꮿ(S, ᐄ). The right (left) translate of f by
s ∈ S is the function RS f (LS f ) such that RS f (t) = f (ts) and (LS f (t) = f (st))
for all t ∈ S. The function f is said to be weakly almost periodic if its right
orbit RS f = {RS f : s ∈ S} is relatively weakly compact in Ꮿ(S, ᐄ). We denote
by ᐃᏭᏼ(S, ᐄ) all such functions.
In the case that ᐄ = C, the complex number field, we will omit ᐄ from our
notations and write, for example, Ꮿ(S) for Ꮿ(S, C).
Recently, some authors have investigated ᐃᏭᏼ(S, ᐄ) and exploited its applications in many areas [1, 2, 4, 5, 6, 7, 8, 9]. However, some questions remain
unsolved. For example, [2, Theorems 4.2.3 and 4.2.6] give a number of equivalent properties for a function f ∈ Ꮿ(S) to be weakly almost periodic. It is natural to ask if the similar equivalent properties are true for a function in Ꮿ(S, ᐄ).
In this paper, we investigate these problems and give positive answers.
It is shown in [4, Proposition 2.8] that the equivalence of relative weak compactness for RS f and the left orbit LS f = {LS f : s ∈ S} holds if S admits an
identity and the range f (S) is relative compact in ᐄ. We will give an example at
the end of the paper to show that the assumptions both on S and on f (S) are
not essential to get the equivalence. We will show the equivalence under the
assumption that f (S) is relatively weakly compact. At the same time, we characterize a vector-valued weakly almost periodic function by giving it as many
equivalent properties as a scalar-valued weakly almost periodic function has.
We will not assume that a semitopological semigroup S admits an identity. In
fact, if S has an identity, we can drop the condition f (S), being either relative
norm compact or relatively weakly compact (see Remark 3.6(b)).
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CHUANYI ZHANG
2. Vector-valued means. To show the main results of the paper in Section 3,
we need some facts of vector-valued means. Unless otherwise mentioned, all
the results of this section come from [8, Sections 2 and 3].
Let ᐆ, ᐅ be two normed linear spaces and let ᐅ∗ be the dual space of ᐅ. Let
ᏸ(ᐆ, ᐅ) be the space of bounded linear operators from ᐆ to ᐅ. With the norm
topology, ᏸ(ᐆ, ᐅ) is a Banach space. We can also furnish ᏸ(ᐆ, ᐅ) with the following two topologies, both of them make ᏸ(ᐆ, ᐅ) a locally convex topological
space [3, VI.1.2, VI.1.3]:
(1) the strong operator topology τs , which is the weakest topology of ᏸ(ᐆ, ᐅ)
relative to which the mapping µ → µ(z) : ᏸ(ᐆ, ᐅ) → ᐅ, is continuous for
each z ∈ ᐆ;
(2) the weak operator topology τw , which is the weakest topology of ᏸ(ᐆ, ᐅ)
relative to which the mapping µ → y ∗ [µ(z)] : ᏸ(ᐆ, ᐅ) → C, is continuous
for each z ∈ ᐆ and y ∗ ∈ ᐅ∗ .
For ᏸ(ᐄ, ᐅ∗ ), we have the following topology that also makes ᏸ(ᐆ, ᐅ∗ ) a locally
convex topological space [3, page 476]:
(3) the weak∗ operator topology τw ∗ , which is the weakest topology of
ᏸ(ᐆ, ᐅ∗ ) relative to which the mapping µ → [µ(z)](y) : ᏸ(ᐆ, ᐅ∗ ) → C,
is continuous for each z ∈ ᐆ and y ∈ ᐅ.
Let S be a nonempty set, let ᐄ be a Banach space, and let Ꮾ(S, ᐄ) be the space
of bounded functions from S to ᐄ with supremum norm. Let Ꮽ be a subspace
of Ꮾ(S, ᐄ) containing the constant functions.
Definition 2.1. A mean µ on Ꮽ is a linear operator from Ꮽ to ᐄ such that
µ(f ) ∈ co f (S) for all f ∈ Ꮽ, denoted by M(Ꮽ) of all means on Ꮽ.
We define the evaluation mapping : S → M(Ꮽ) as follows: for s ∈ S, (s)f =
f (s), f ∈ Ꮽ. The following proposition comes from [9, Propositions 1.5 and
1.6].
Proposition 2.2. Let Ꮽ be a subspace of Ꮾ(S, ᐄ) containing the constant
functions. Then for both τs and τw , M(Ꮽ) is convex and closed in ᏸ(Ꮽ, ᐄ), and
co((S)) is dense in M(Ꮽ). Furthermore, if Ꮽ is such that f (S) is relatively
weakly compact in ᐄ for all f ∈ Ꮽ, then M(Ꮽ) is τw -compact.
We embed ᐄ into its double dual space ᐄ∗∗ canonically and let ι(ᐄ) denote
its canonical image in ᐄ∗∗ ; similarly, we embed f (S) into ᐄ∗∗ for every f ∈ Ꮽ
and get a subspace ι(Ꮽ) of Ꮾ(S, ι(ᐄ)). A function of Ꮽ may be regarded as
a function of ι(Ꮽ), and vice versa. Replacing Ꮽ and ᐄ by ι(Ꮽ) and ᐄ∗∗ in
Definition 2.1, respectively, we get M(ι(Ꮽ)). A mean of M(Ꮽ) may be regarded
as a mean of M(ι(Ꮽ)), and vice versa. This leads to the following more general
definition of means.
Definition 2.3. Let Ꮽ be a subspace of Ꮾ(S, ᐄ∗∗ ). A linear map µ : Ꮽ → ᐄ∗∗
∗
is called a w ∗ mean on Ꮽ provided µ(f ) ∈ cow f (S), for all f ∈ Ꮽ, where
w ∗ stands for the weak∗ topology σ (ᐄ∗∗ , ᐄ∗ ). Denote by w ∗ M(Ꮽ) the set of
VECTOR-VALUED W.A.P. FUNCTIONS
297
all w ∗ means on Ꮽ. In the case that Ꮽ ⊂ Ꮾ(S, ᐄ), we define w ∗ M(Ꮽ) to be
w ∗ M(ι(Ꮽ)).
Both Definitions 2.1 and 2.3 will reduce to the definition of a scalar-valued
mean when ᐄ = C [1, 2.1.2].
Proposition 2.4. Let Ꮽ be a linear subspace of Ꮾ(S, ᐄ∗∗ ). Then for τw ∗
(1) w ∗ M(Ꮽ) is convex and compact;
(2) let be the evaluation map S → w ∗ M(Ꮽ), then co((S)) is dense in
w ∗ M(Ꮽ).
Proposition 2.5. Every member of w ∗ M(Ꮽ) can be extended to a member
of w ∗ M(Ꮾ(S, ᐄ∗∗ )).
We call the scalar-valued function space
Ᏺ = sp
f (·) x ∗ : x ∗ ∈ ᐄ∗ , f ∈ Ꮽ
(2.1)
generated space of Ꮽ.
Proposition 2.6. Let Ꮽ be a linear subspace of Ꮾ(S, ᐄ∗∗ ) containing the
constant functions, and let Ᏺ be its generated space. Then, there is an isometric
τw ∗ -σ (Ᏺ∗ , Ᏺ) homeomorphism µ → ϕµ : w ∗ M(Ꮽ) → M(Ᏺ) such that
µ(f ) x ∗ = ϕµ f (·) x ∗
f ∈ Ꮽ, x ∗ ∈ ᐄ ∗ .
(2.2)
3. Main results. A nonempty set S that is a semigroup and also a topological
space is called a semitopological semigroup provided that the maps s → ts and
s → st from S to S are continuous for all t ∈ S. Let S be such a set, and let Ꮽ
be a subspace of Ꮿ(S, ᐄ). We say Ꮽ is right (resp., left) translation invariant if
RS Ꮽ = {RS f : s ∈ S, f ∈ Ꮽ} ⊂ Ꮽ (resp., LS Ꮽ = {LS f : s ∈ S, f ∈ Ꮽ} ⊂ Ꮽ). We
say Ꮽ is translation invariant if it is both right and left translation invariant.
Let Ꮽ be a translation invariant subspace of Ꮿ(S, ᐄ). For µ ∈ M(Ꮽ), define
Tµ : Ꮽ → Ꮾ(S, ᐄ) by
Tµ f (s) = µ Ls f
(f ∈ Ꮽ, s ∈ S)
(3.1)
(f ∈ Ꮽ, s ∈ S).
(3.2)
and Uµ : Ꮽ → Ꮾ(S, ᐄ) by
Uµ f (s) = µ Rs f
We call Tµ (Uµ ) left (right) introversion operator determined by µ. We will say
that Ꮽ is left (right) introverted if Tµ Ꮽ ⊂ Ꮽ (Uµ Ꮽ ⊂ Ꮽ) for all µ ∈ M(Ꮽ). We will
say that Ꮽ is introverted if it is both left and right introverted.
Similarly, we define an introversion operator from Ꮽ to Ꮾ(S, ᐄ∗∗ ) if Ꮽ is a
translation invariant subspace of Ꮾ(S, ᐄ∗∗ ) and µ ∈ w ∗ M(Ꮽ).
To show Theorem 3.2, we need the following proposition that characterizes
weak almost periodicity of a function in Ꮿ(S).
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CHUANYI ZHANG
Proposition 3.1 [2, 4.2.6]. Let S be a semitopological semigroup, let f ∈
Ꮿ(S), and let Ᏺ = ᐃᏭᏼ(S). Then, the following statements are equivalent:
(1) f ∈ ᐃᏭᏼ(S);
(2) LS f is relatively weakly compact in Ꮿ(S);
(3) the mapping ϕ → Tϕ f : M(Ᏺ) → Ꮾ(S) is σ (Ᏺ∗ , Ᏺ) → σ (Ꮾ(S), Ꮾ(S)∗ )
continuous;
(4) the mapping ϕ → Uϕ f : M(Ᏺ) → Ꮾ(S) is σ (Ᏺ∗ , Ᏺ) → σ (Ꮾ(S), Ꮾ(S)∗ )
continuous;
(5) for all ϕ,ψ ∈ M(Ᏺ), ϕ(Tψ f ) = ψ(Uϕ f ).
We will generalize Proposition 3.1 from scalar-valued function to vectorvalued function in the next theorem. We will use some results of the previous
section to show the theorem. To make notations short, we let
ι(Ꮿ) = Ꮿ S, ι(ᐄ) ,
ι(Ꮾ) = Ꮾ S, ι(ᐄ) ,
Ꮾ = Ꮾ S, ᐄ∗∗ .
(3.3)
As in the paragraph before Definition 2.3, ι(ᐄ) is the canonical image of ᐄ in
ᐄ∗∗ ; an f ∈ Ꮾ(S, ᐄ) and its corresponding function in ι(Ꮾ) will be regarded as
same function.
Note that both Ꮾ and ι(Ꮾ) have the same generated space Ꮾ(S), the space
of bounded scalar-valued functions on S.
Theorem 3.2. Let S be a semitopological semigroup and let ᐄ be a Banach
space. Let Ꮽ = ᐃᏭᏼ(S, ᐄ) and let f ∈ Ꮿ(S, ᐄ) be such that f (S) is relatively
weakly compact in ᐄ. Then the following statements are equivalent:
(1) f ∈ Ꮽ, that is, RS f is relatively weakly compact in Ꮿ(S, ᐄ);
(2) LS f is relatively weakly compact in Ꮿ(S, ᐄ);
(3) the mapping µ → Tµ f : w ∗ M(Ꮾ) → ι(Ꮾ) is τw ∗ -σ (ι(Ꮾ), ι(Ꮾ)∗ ) continuous;
(4) the mapping µ → Uµ f : w ∗ M(Ꮾ) → ι(Ꮾ) is τw ∗ -σ (ι(Ꮾ), ι(Ꮾ)∗ ) continuous;
(5) for all µ, ν ∈ w ∗ M(Ꮾ),
µ Tν f = ν Uµ f .
(3.4)
Proof. Since f (S) is relatively weakly compact in ᐄ, the functions Tµ f and
Uµ f are in ι(Ꮾ) for all µ ∈ w ∗ M(Ꮾ).
Let and be the evaluation mappings on Ꮾ and Ꮾ(S), respectively. Let
B = ccoτw ∗ (S), and let BᏮ(S)∗ be the unit ball of Ꮾ(S)∗ . By [2, 2.1.14], BᏮ(S)∗ =
∗
ccow (S), where w ∗ stands for σ (Ꮾ(S)∗ , Ꮾ(S)). It follows from Propositions
2.4(2) and 2.6 that B and BᏮ(S)∗ are τw ∗ -σ (Ꮾ(S)∗ , Ꮾ(S)) homeomorphic.
Define V : B → Ꮾ by
V (µ) = Tµ f .
(3.5)
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299
V is continuous from τw ∗ to the weak∗ pointwise convergence topology P . For,
if {µα } ⊂ B and µ ∈ B are such that µα → µ in τw ∗ , then
∗ ∗ → µ LS f
x
x
Tµα f (s) x ∗ = µα LS f
∗ ∗
∗
x ∈ᐄ , s∈S .
= Tµ f (s) x
(3.6)
Since B is τw ∗ -compact (Proposition 2.4(1)), we have
V (B) = cco RS f ,
(3.7)
where closure in ι(Ꮾ) is taken in the weak∗ pointwise convergence topology P .
Now, we show that (1) implies (3).
By the Krein-Smulian theorem [2, Theorem A.10], cco(RS f ) is relatively
weakly compact in ι(Ꮿ), which in view of (3.7) implies that V (B) is σ (ι(Ꮿ),
ι(Ꮿ)∗ )-closure of cco(RS f ) in ι(Ꮿ) and that V (B) is σ (ι(Ꮿ), ι(Ꮿ)∗ )-compact.
Therefore, the weak topology σ (ι(Ꮿ), ι(Ꮿ)∗ ) and the topology P coincide on
V (B). So, V is τw ∗ -σ (ι(Ꮾ), ι(Ꮾ)∗ ) continuous on B.
To show (3), we define, for Φ ∈ ι(Ꮾ)∗ , the linear functional Φ ◦T : Ꮾ(S)∗ → C
by
Φ ◦ T ϕµ = Φ Tµ f ,
(3.8)
where µ and ϕµ are as in (2.2). It follows from the τw ∗ -σ (ι(Ꮾ), ι(Ꮾ)∗ ) continuity on B of V that Φ ◦ T is σ (Ꮾ(S)∗ , Ꮾ(S)) continuous on BᏮ(S)∗ . By
Grothendieck’s completeness theorem [2, Proposition A.8], Φ ◦ T is σ (Ꮾ(S)∗ ,
Ꮾ(S)) continuous on Ꮾ(S)∗ . Now, we claim that (3) holds. For, if µα and µ
of w ∗ M(Ꮾ) are such that µα → µ in τw ∗ , then ϕµα → ϕµ in σ (Ꮾ(S)∗ , Ꮾ(S))
(Proposition 2.6), and therefore
Φ Tµα f = Φ ◦ T ϕµα → Φ ◦ T ϕµ = Φ Tµ f .
(3.9)
Since Φ is arbitrary in ι(Ꮾ)∗ , we have Tµα f → Tµ f in σ (ι(Ꮾ), ι(Ꮾ)∗ ). Thus (3)
holds.
By Proposition 2.4(1), B is τw ∗ -compact. If (3) holds, then it follows from
(3.7) that RS f is relatively weakly compact in ι(Ꮾ). Thus (1) holds. So, (1) and
(3) are equivalent.
Similarly, we show that (2) and (4) are equivalent. Next, we show that (1) and
(3) imply (5).
Let µ, ν ∈ w ∗ M(Ꮾ) and ϕν , ϕµ ∈ M(Ꮾ(S)) be as in (2.2). Since f is in
ᐃᏭᏼ(S, ᐄ), f (·)(x ∗ ) is in ᐃᏭᏼ(S) for all x ∗ ∈ ᐄ∗ . By Proposition 3.1(5),
ϕν Tϕµ f (·) x ∗
= ϕµ Uϕν f (·) x ∗
.
(3.10)
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CHUANYI ZHANG
It follows from (2.2) and (3.10) that
∗ x = ϕν Tϕµ f (·) x ∗
ν Tµ f
= ϕµ Uϕν f (·) x ∗
∗ x .
= µ Uν f
(3.11)
Since x ∗ ∈ ᐄ∗ is arbitrary, we have
ν Tµ f = µ Uν f
µ, ν ∈ w ∗ M(Ꮾ) .
(3.12)
Thus (5) holds.
Similarly, we show that (2) and (4) imply (5).
For x ∗ ∈ ᐄ∗ and µ ∈ w ∗ M(ι(Ꮾ)), define µ ◦ x ∗ : ι(Ꮾ) → C by
h ∈ ι(Ꮾ) .
µ ◦ x ∗ (h) = µ(h) x ∗
(3.13)
Then, µ ◦x ∗ ∈ ι(Ꮾ)∗ . Let D = {µ ◦x ∗ : x ∗ ∈ ᐄ∗ , x ∗ = 1, µ ∈ w ∗ M(ι(Ꮾ))}. By
∗
the separation theorem, we show that ccow D = Bι(Ꮾ)∗ , the unit ball of ι(Ꮾ)∗ ,
where w ∗ stands for σ (ι(Ꮾ)∗ , ι(Ꮾ)).
To show that (5) implies (4), we need to show that the mapping µ → Uµ f :
B = ccoτw ∗ (S) → ι(Ꮾ) is τw ∗ -σ (ι(Ꮾ), ι(Ꮾ)∗ ) continuous. That is, we need to
show that if µα ⊂ B and µ ∈ B are such that µα → µ in τw ∗ , and if F ∈ Bι(Ꮾ)∗ ,
then
F Uµα f → F Uµ f .
(3.14)
∗
Note that ccow D is the unit ball of ι(Ꮾ)∗ . For ν ◦ x ∗ ∈ cco D, we have
ν ◦ x ∗ Uµα f → ν ◦ x ∗ Uµ f ,
(3.15)
because it follows from (3.4) and (3.13) that
∗ ν ◦ x ∗ Uµα f = ν Uµα f
x
∗ ∗ = µα T ν f
x
x
→ µ Tν f
∗ = ν Uµ f
x
∗
= ν ◦ x Uµ f .
(3.16)
Let A = {Uµ f : µ ∈ B}. Then, A is a bounded subset of ι(Ꮾ). As in the discussion for Note (ii) and (iii) before Theorem 3 of [5], we regard cco D as a set
of bounded function on A, that is, a subset of Ꮾ(A). Then, the weak∗ closure
VECTOR-VALUED W.A.P. FUNCTIONS
301
w∗
cco D in ι(Ꮾ)∗ is contained in the weak closure ccow D in Ꮾ(A). Since ccow D
∗
is also norm closed in Ꮾ(A), for F ∈ ccow D ⊂ ccow D, we have a sequence of
∗
{νn ◦ xn
} of cco D such that
F (h) − νn ◦ x ∗ (h) → 0
n
(3.17)
uniformly in h ∈ A. Since
F Uµ
αf
∗
− F Uµ f ≤ F Uµα f − νn ◦ xn
Uµα f ∗
∗
Uµα f − νn ◦ xn
Uµ f + νn ◦ xn
∗
+ νn ◦ x n
Uµ f − F Uµ f ,
(3.18)
now (3.14) is a consequence of (3.15) and (3.17).
Similarly, we show that (5) implies (3). The proof is complete.
Corollary 3.3. Let S, ᐄ, and Ꮽ be as in Theorem 3.2. Let f ∈ Ꮽ. Then,
Tµ f ∈ Ꮽ for all µ ∈ M(Ꮽ). Furthermore, if f (S) is relatively weakly compact in
ᐄ, then Uµ f ∈ Ꮽ for all µ ∈ M(Ꮽ).
Proof. As in the proof of (1) implying (3) of Theorem 3.2, we show the first
statement. If f (S) is relatively weakly compact in ᐄ, then by the theorem, LS f
is relatively weakly compact in Ꮿ(S, ᐄ). Note that this time f is in Ꮽ. We show
the second statement as in the proof of (2) implying (4).
For every x ∗ ∈ ᐄ∗ , x ∗ = 1, and s ∈ S, define x ∗ ◦ s : Ꮿ(S, ᐄ) → C, by
x ∗ ◦ s(f ) = x ∗ f (s)
f ∈ Ꮿ(S, ᐄ) .
(3.19)
Then, x ∗ ◦ s ∈ Ꮿ(S, ᐄ)∗ , the dual space of Ꮿ(S, ᐄ). Set E = {x ∗ ◦ s : x ∗ ∈
w∗
ᐄ∗ , x ∗ = 1, s ∈ S}. Let B = E , where w ∗ stands for the weak∗ topology
σ (Ꮿ(S, ᐄ)∗ , Ꮿ(S, ᐄ)). Then, B is weak∗ compact.
For every f ∈ Ꮿ(S, ᐄ), define fˆ : E → C, by
fˆ x ∗ ◦ s = x ∗ f (s)
∗
x ◦s ∈ E .
(3.20)
We extend fˆ from E to B continuously. So we have fˆ ∈ Ꮿ(B).
∗
∗
Obviously, we have R
t f for t ∈ S and Rt f , that is, Rt f (x ◦s) = x [Rt f (s)].
ˆ
The mapping f → f : Ꮿ(S, ᐄ) → Ꮿ(B) is one to one and linear isometric. The
space Ꮿ(B) is a Banach space with norm topology. In the next theorem, we will
also equip Ꮿ(B) with the P -topology, the pointwise convergence topology.
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CHUANYI ZHANG
Theorem 3.4. Let S be a semitopological semigroup, and let f ∈ Ꮿ(S, ᐄ)
such that f (S) is relatively weakly compact in ᐄ. Then, the following statements
are equivalent:
(1) f ∈ ᐃᏭᏼ(S, ᐄ);
(2) LS f is relatively weakly compact in Ꮿ(S, ᐄ);
(3) R
S f is relatively compact in Ꮿ(B) in the P -topology;
(4) L
S f is relatively compact in Ꮿ(B) in the P -topology;
∗
∗
(5) limm→∞ limn→∞ xm
f (sm tn ) = limn→∞ limm→∞ xm
f (sm tn ), whenever
∗
∗
∗
{xm } ⊂ ᐄ , x = 1, {tn }, {sm } ⊂ S and all the limits exist;
∗
∗
f (tn sm ) = limn→∞ limm→∞ xm
f (tn sm ), whenever
(6) limm→∞ limn→∞ xm
∗
∗
∗
{xm } ⊂ ᐄ , x = 1, {tn }, {sm } ⊂ S and all the limits exist.
Proof. The equivalence of (1) and (2) comes from Theorem 3.2.
If (1) holds, then RS f is relatively weakly compact in Ꮿ(S, ᐄ). Since the map
ping f → fˆ is one to one and linear isometric, R
S f is relatively weakly compact
in Ꮿ(B). So, RS f is relatively compact in Ꮿ(B) in P -topology. Thus (3) holds.
∗
∗
Now, we show (3) implies (5). Let {xm
} ⊂ ᐄ∗ , xm
= 1, and {tn }, {sm } ⊂ S
be sequences such that the iterated limits
∗
f s m tn ,
lim lim xm
∗
f s m tn
lim lim xm
m→∞ n→∞
n→∞ m→∞
(3.21)
∗
exist. Thus, we have {R
tn f } ⊂ Ꮿ(B) and {xm ◦ sm } ⊂ E. Let ĝ ∈ Ꮿ(B) be P ∗
topological cluster point of {R
tn f }, and let y ∈ B be cluster point of {xm ◦sm }.
Then,
∗
∗
lim lim R
tn f xm ◦ sm = lim ĝ xm ◦ sm
m→∞ n→∞
m→∞
= ĝ(y) = lim R
tn f (y)
n→∞
(3.22)
∗
= lim lim R
tn f xm ◦ sm .
n→∞ m→∞
Note that
∗
∗
lim lim R
tn f xm ◦ sm = lim lim xm f sm tn ,
m→∞ n→∞
m→∞ n→∞
n→∞ m→∞
n→∞ m→∞
∗
∗
lim lim R
tn f xm ◦ sm = lim lim xm f sm tn .
(3.23)
Therefore, (5) holds.
That (5) implies (1) is a consequence of Grothendieck’s double theorem [2,
A.5]. Similarly, we show the equivalence among (2), (4), and (6). The proof is
complete.
Example 3.5. Let S = N = {1, 2, . . .}, the semigroup of natural numbers.
Let lp , 1 < p < ∞ be the usual sequence spaces with basis {en }∞
n=1 , where
with
x
=
1
if
k
=
n
and
x
=
0
otherwise.
Define
f
:
N
→ lp by
en = {xk }∞
k
k
k=1
f (n) = en
(n ∈ N).
(3.24)
VECTOR-VALUED W.A.P. FUNCTIONS
303
Since N is abelian, RS f = LS f . For any subsequences {mi }, {nk } of S, and {ϕi }
of lq with ϕ 1, where 1/p +1/q = 1 and the norm being taken in the space
lq , we have that
lim lim ϕi f mi + nk = lim lim ϕi emi +nk = lim lim ϕi,mi +nk = 0,
i
k
i
k
i
k
k
i
k
i
k
i
lim lim ϕi f mi + nk = lim lim ϕi emi +nk = lim lim ϕi,mi +nk = 0,
(3.25)
where ϕi,mi +nk is the (mi +nk )th component of ϕi . Therefore, by Theorem 3.4
we have f ∈ ᐃᏭᏼ(N, lp ). However, f (S) is not relatively norm compact but
relatively weakly compact. Since RS f = LS f , the two orbits of f are all relatively
weakly compact. We note that N does not admit an identity.
Remark 3.6. (a) The equivalence of (1) and (5) of Theorem 2.4 appeared in
[5, Theorem 6]; though it was assumed that S admits an identity, the proof
of Theorem 6 did not use the identity. (b) In both Theorems 3.2 and 3.4, we
assume that the range of f (S) is relatively weakly compact to show that the
relative weak compactness of LS f is equivalent to that of RS f . We do not
know if the condition of f (S) is essential. We showed in [8, Corollary 8.4] that
if S admits an identity, then f (S) is relatively weakly compact in ᐄ for all
f ∈ ᐃᏭᏼ(S, ᐄ). Of course, if ᐄ is reflexive, then any bounded function has a
relatively weakly compact range. (c) From the proof of Theorem 3.4, we see that
to get the equivalence among (1), (3), and (5), it does not need the condition
f (S) being relatively weakly compact, neither does the equivalence among (2),
(4), and (6).
Acknowledgments. The research is supported in part by the National
Science Foundations (NSF) of China and Harbin Institute of Technology (HIT).
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Chuanyi Zhang: Department of Mathematics, Harbin Institute of Technology, Harbin
150001, China
E-mail address: czhang@hope.hit.edu.cn